qq-Partition Algebra Combinatorics

We compute the dimension d_{n,r}(q) = \dim(\IR_q^r) of the defining module \IR_q^r for the $q$-partition algebra. This module comes from $r$-iterations of Harish-Chandra restriction and induction on \GL_n(\FF_q). This dimension is a polynomial in $q$ that specializes as $d_{n,r}(1) = n^r$ and $d_{n,r}(0) = B(r)$, the $r$th Bell number. We compute $d_{n,r}(q)$ in two ways. The first is purely combinatorial. We show that $d_{n,r}(q) = \sum_\lambda f^\lambda(q) m_r^\lambda$, where $f^\lambda(q)$ is the $q$-hook number and $m_r^\lambda$ is the number of $r$-vacillating tableaux. Using a Schensted bijection, we write this as a sum over integer sequences which, when $q$-counted by inverse major index, gives $d_{n,r}(q)$. The second way is algebraic. We find a basis of \IR_q^r that is indexed by $n$-restricted $q$-set partitions of $\{1,..., r\}$, and we show that there are $d_{n,r}(q)$ of these.Comment: Introduction rewritten and minor mistakes correcte

On the Limiting Vacillating Tableaux for Integer Sequences

A fundamental identity in the representation theory of the partition algeba is $n^k = \sum_{\lambda} f^\lambda m_k^\lambda$ for $n \geq 2k$, where $\lambda$ ranges over integer partitions of $n$, $f^\lambda$ is the number of standard Young tableaux of shape $\lambda$, and $m_k^\lambda$ is the number of vacillating tableaux of shape $\lambda$ and length $2k$. Using a combination of RSK insertion and jeu de taquin, Halverson and Lewandowski constructed a bijection $DI_n^k$ that maps each integer sequence in $[n]^k$ to a pair consisting of a standard Young tableau and a vacillating tableau. In this paper, we show that for a given integer sequence $\boldsymbol{i}$, when $n$ is sufficiently large, the vacillating tableaux determined by $DI_n^k(\boldsymbol{i})$ become stable when $n \rightarrow \infty$; the limit is called the limiting vacillating tableau for $\boldsymbol{i}$. We give a characterization of the set of limiting vacillating tableaux and presents explicit formulas that enumerate those vacillating tableaux

Combinatorics of Oscillating Tableaux

In this paper, we first introduce the RSK algorithm, which gives a correspondence between integer sequences and standard tableaux. Then we introduce Schensted’s theorem and Greene’s theorem that describe how the shape of the standard tableau is determined by the sequence. We list four different bijections constructed by using the RSK insertion. The first one is a bijection between vacillating tableaux and pairs (P, T), where P is a set of ordered pairs and T is a standard tableau. The second one is a bijection between set partitions of [n] and vacillating tableaux. The third one is about partial matchings and up-down tableaux and the final one is from sequences to pairs (T, P), where T is still a standard tableau and P is a special oscillating tableau. We analyze some combinatorial statistics via these bijections

Combinatorics of Oscillating Tableaux

In this paper, we first introduce the RSK algorithm, which gives a correspondence between integer sequences and standard tableaux. Then we introduce Schensted’s theorem and Greene’s theorem that describe how the shape of the standard tableau is determined by the sequence. We list four different bijections constructed by using the RSK insertion. The first one is a bijection between vacillating tableaux and pairs (P, T), where P is a set of ordered pairs and T is a standard tableau. The second one is a bijection between set partitions of [n] and vacillating tableaux. The third one is about partial matchings and up-down tableaux and the final one is from sequences to pairs (T, P), where T is still a standard tableau and P is a special oscillating tableau. We analyze some combinatorial statistics via these bijections

Motivating a q-Version of the Embedding Between the Categories Heis and Par(t)

Within this work, assuming an undergraduate background, we will introduce the material necessary to understand five categories and their relationships. These include the Heisenberg category&nbsp;Heis, the partition category&nbsp;Par(t), the category of&nbsp;kSn-modules&nbsp;Sn-mod, the&nbsp;q-deformed Heisenberg category&nbsp;Heis(q), and&nbsp;GLn(Fq)-mod. The main aim of the work done in this paper is to motivate a generalization of a particular isomorphism of functors constructed by Likeng and Savage&nbsp;between the non-q-deformed categories&nbsp;Par(n),&nbsp;Heis&uarr;&darr;(n), and&nbsp;Sn-mod&nbsp;to a similar isomorphism between functors on their&nbsp;q- deformed counterparts. We offer a conjectured embedding into&nbsp;Heis&uarr;&darr;(q)&nbsp;and demonstrate which relations of the partition category are respected by the embedding. </div

Diagram monoids and Graham–Houghton graphs: Idempotents and generating sets of ideals

We study the ideals of the partition, Brauer, and Jones monoid, establishing various combinatorial results on generating sets and idempotent generating sets via an analysis of their Graham–Houghton graphs. We show that each proper ideal of the partition monoid Pn is an idempotent generated semigroup, and obtain a formula for the minimal number of elements (and the minimal number of idempotent elements) needed to generate these semigroups. In particular, we show that these two numbers, which are called the rank and idempotent rank (respectively) of the semigroup, are equal to each other, and we characterize the generating sets of this minimal cardinality. We also characterize and enumerate the minimal idempotent generating sets for the largest proper ideal of Pn, which coincides with the singular part of Pn. Analogous results are proved for the ideals of the Brauer and Jones monoids; in each case, the rank and idempotent rank turn out to be equal, and all the minimal generating sets are described. We also show how the rank and idempotent rank results obtained, when applied to the corresponding twisted semigroup algebras (the partition, Brauer, and Temperley–Lieb algebras), allow one to recover formulae for the dimensions of their cell modules (viewed as cellular algebras) which, in the semisimple case, are formulae for the dimensions of the irreducible representations of the algebras. As well as being of algebraic interest, our results relate to several well-studied topics in graph theory including the problem of counting perfect matchings (which relates to the problem of computing permanents of {0,1}-matrices and the theory of Pfaffian orientations), and the problem of finding factorizations of Johnson graphs. Our results also bring together several well-known number sequences such as Stirling, Bell, Catalan and Fibonacci numbers